We study the maximum number of 2xed points of boolean networks with local update function AND–OR. We prove that this number for networks with connected digraph is 2(n−1)=2 for n odd and 2(n−2)=2 + 1 for n even if the digraph has not loops; and 2n−1 + 1 otherwise, where n is the number of nodes of the digraph. We also exhibit some networks reaching these bounds. To obtain these results we constr...

Let G=(V,A) be a digraph. The eccentricity e(u) of a vertex u is the maximum distance from u to any other vertex in G. A vertex v in G is an eccentric vertex of u if the distance from u to v equals e(u). The eccentric digraph ED(G) of a digraph G has the same vertex set as G and has arcs from a vertex v to its eccentric vertices. In this paper we present several results on the eccentric digraph...

An upward planar drawing of a digraph G is a planar drawing of G where every edge is drawn as a simple curve monotone in the vertical direction. A digraph is upward planar if it has an embedding that admits an upward planar drawing. The problem of testing whether a digraph is upward planar is NP-complete. In this paper we give a linear-time algorithm to test the upward planarity of a series-par...

A minimum reversing set of a digraph is a smallest sized set of arcs which when reversed makes the digraph acyclic. We investigate a related issue: Given an acyclic digraph D, what is the size of a smallest tournataent T which has the arc set of D as a minimum reversing set? We show that such a T always exists and define the reversing number ofan acyclic digraph to be the number of vertices in ...

A Cayley digraph G = C(Γ, X) for a group Γ and a generating set X is the digraph with vertex set V (G) = Γ and arcs (g, gx) where g ∈ Γ and x ∈ X. The reverse of C(Γ, X) is the Cayley digraph G−1 = C(Γ, X−1) where X−1 = {x−1;x ∈ X}. We are interested in sufficient conditions for a Cayley digraph not to be isomorphic to its reverse and focus on Cayley digraphs of metacyclic groups with small gen...

In this paper we introduce two polytopes that respect a digraph in the sense that for every vector in the polytope every component corresponds to a node and is at least equal to the component corresponding to each successor of this node. The sharing polytope is the set of all elements from the unit simplex that respect the digraph. The fuzzy polytope is the set of all elements of the unit cube ...

A minimum feedback arc set of a digraph is a smallest sized set of arcs that when reversed makes the resulting digraph acyclic. Given an acyclic digraph D, we seek a smallest sized tournament T that has D as a minimum feedback arc set. The reversing number of a digraph was defined by Barthélemy et. al. to be r(D) = |V (T )| − |V (D)|. We will completely determine the reversing number for a disj...

The support of a matrix M is the (0, 1)-matrix with ij-th entry equal to 1 if the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose adjacency matrix is the support of M is said to be the digraph of M . This paper observes some structural properties of digraphs and Cayley digraphs, of unitary matrices. We prove that a group generated by two elements has a set of generato...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A kernel N of D is an independent set of vertices such that for every w∈V (D) − N there exists an arc from w to N . A digraph D is called right-pretransitive (resp. left-pretransitive) when (u; v)∈A(D) and (v; w)∈A(D) implies (u; w)∈A(D) or (w; v)∈A(D) (resp. (u; v)∈A(D) and (v; w)∈A(D) implies (u; ...